Resolution-Exact Planner for Non-Crossing 2-Link Robot Zhongdi Luo and Chee K. Yap Department of Computer Science Courant Institute, NYU New York, NY 10012, USA {zl562,yap}@cs.nyu.edu Abstract— We consider the motion planning problem for a 2-link robot amidst polygonal obstacles. The two links are normally allowed to cross each other, but in this paper, we introduce the non-crossing version of this robot. This 4DOF configuration space is novel and interesting. Using the recently introduced algorithmic framework of Soft Subdivision Search (SSS), we design a resolution-exact planner for this robot. We introduce a new data structure for representing boxes and doing subdivision in this non-crossing configuration space. Our implementation achieved real-time performance for a suite of non-trivial obstacle sets. I. I NTRODUCTION Motion planning is one of the key topics of robotics [7], [3]. The main approach to motion planning for almost two decades now is probabilistic sampling such as PRM [6]. In the plane, the simplest example of a flexible or non- rigid robot is the 2-link robot, R 2 = R 2 (ℓ 1 ,ℓ 2 ), with links of positive lengths ℓ 1 and ℓ 2 . The two links are connected through a rotational joint A 0 called the robot origin as illustrated in Figure 1(a). The 2-link robot is in the intersection of two well-known families of link robots: chain robots and spider robots (Figure 1(b,c)). See [9] for a definition of link robots; there we also discuss link robots with thickness. (b) Chain Robot (c) Spider Robot A0 A2 A5 A0 (a) R2 θ2 θ1 A0 A2 A1 A1 A4 A3 Fig. 1: Link Robots We address the following phenomena: in the screen shot of Figure 2, we show two configurations of the 2-link robot inside an (inverted) T-room environment. Let α (respectively, β) be the start (goal) configuration as indicated 1 by the double (single) circle. There are obvious paths from α to β whereby the robot origin moves directly from the start to goal positions and simultaneously, the link angles monotonically 1 Our images have color: the double circle is cyan and single circle is magenta. The robot links are colored blue (ℓ 1 ) and red (ℓ 2 ), respectively. adjust themselves, as illustrated in Figure 4(a). However, such paths require the two links to cross each other. To achieve a “non-crossing” path from α to β, the robot origin must initially move away from the goal configuration towards the T-junction first, in order to maneuver the two links into the appropriate relative order before it can move toward the goal configuration. Such a non-crossing path is shown in Figure 4(b). We find such paths with a subdivision algo- rithm; Figure 3 illustrates the subdivision of the underlying configuration space (the scheme is explained below). Fig. 2: T-Room: start and goal configurations This paper shows how to construct a practical and theoretically-sound planner for a non-crossing 2-link robot. To our knowledge, this non-crossing configuration space has not been studied before. Our planner may (but not always) suffer a loss of efficiency when compared to the self-crossing 2-link planner (see [9]). Nevertheless, it gives real-time per- formance for a variety of non-trivial obstacle environments such as illustrated in Figure 5 (200 randomly generated triangles), Figure 6(a) (Double Bug-trap (cf. [p. 181, Fig- ure 5.13][7]), Figure 6(b) (Maze). Unlike sampling based planners, we do not need any pre-processing arguments, and no special techniques are needed to address the halting or narrow-passage problem. For example, Figure 5(a) is an environment with 200 randomly generated triangles, and a path is found with ε =4. If we use ε =5, then it returns NO- PATH as shown in Figure 5(b). This NO-PATH declaration